Question: In the book Continued Fractions, A. Ya. Khinchin 1964 ( $-$?), Chapter II: The Representation of Numbers By Continued Fractions page 19 he writes
"Continued fractions have an undeniable and considerable advantage
over systematic ( and in particular , decimal ) fractions$\ldots$
Since a systematic fraction is connected with a certain system of
calculations, it therefore unavoidably reflects, not so much the
absolute properties of the number it represents as its relationship to
that particular system of calculation. Continued fractions on the
other hand are not connected with any system of calculations; they
produce in pure form the properties of the number that they
represent."
What exactly is meant by this statement ? And how are one fractions connected to a system of calculations and the others not. And what system of calculation might he be referring to; standard arithmetic $+,-,\times, \div$ ?
Best Answer
Khinchin is referring to the usual way of representing real numbers, as sequences of base-$b$ digits: $$x = \sum_{i=0}^\infty a_ib^i\qquad\text{$0≤a_i<b$ when $i≥1$}$$
This system is familiar from grade school. It is fairly simple in some ways, but it has warts, and the warts are in odd and mathematically inconvenient places. For example, when $b=10$, even a simple everyday number like $\frac13$ has no finite representation, only an infinite one, whereas a complicated number like $\frac{142857}{781250}$ has a simple finite representation. Why? Because of the non-obvious fact that there exists an integer $k$ for which $781250$ divides $10^k$, but no analogous $k$ exists for $3$. (“Not so much the absolute properties of the number, as its relationship to [the base-$10$ system].”)
Every real number can be represented, but some numbers have more than one representation (for example, $\frac12 = 0.5000\ldots = 0.4999\ldots$) and which ones have multiple representations is again tied up with properties of the number $10$ that are not usually of any relevance.
The addition and multiplication operations on systematic fractions seem relatively straightforward… until you ask how to calculate $\frac16+\frac16$, when all sorts of difficulties start to intrude. The usual algorithm you learned in school asks you to start with the rightmost digits, but $\frac16$ has no rightmost digits. So you start with the leftmost digits, which are $1$ and $1$, and add them to get $2$… which is wrong, it should be $3$, and you have to correct it later. And in general, the correction might not become apparent until an arbitrarily long time later, or not at all. (When adding $0.1666\ldots$ to $0.3333\ldots$ you never do find out for sure whether the tenths-place digit is $4$ or $5$!)
Continued fractions make up for these problems. They are base-agnostic: all rational numbers have exactly two (trivially different) representations; all irrational numbers have exactly one representation. They are only a little more difficult to compare than systematic numerals. And (unknown in Khinchin's time) there are relatively simple algorithms for adding, multiplying, and dividing them. (The “much later correction” problem still arises, but only in examples where you would expect it to arise, like $\sqrt2\cdot \sqrt 2$, and not in problems that should be simple, like $\frac16 + \frac1{3}$.)